We recall here the notions and results of [1]} and [2]} about d-critical structures and the Donaldson-Thomas sheaf. Joyce and its collaborators have developed the theory of d-critical schemes and build the perverse sheaf on it using the Zariski topology, mainly in order to define also motivic Donaldson-Thomas invariants, which can be glued in the Zariski topology, but not in the étale topology. But because coherent sheaves and perverse sheaves glue in the étale topology, and the perverse sheaf of vanishing cycles transforms naturally under étale maps, the same formalism can be developed in the étale topology. We will then work in the étale topology, and consider d-critical algebraic spaces.
